March 2024 Singular kinetic equations and applications
Zimo Hao, Xicheng Zhang, Rongchan Zhu, Xiangchan Zhu
Author Affiliations +
Ann. Probab. 52(2): 576-657 (March 2024). DOI: 10.1214/23-AOP1666


In this paper we study singular kinetic equations on R2d by the paracontrolled distribution method introduced in Gubinelli, Imkeller and Perkowski (Forum Math. Pi 3 (2015) e6–75). We first develop paracontrolled calculus in the kinetic setting and use it to establish the global well-posedness for the linear singular kinetic equations under the assumptions that the products of singular terms are well defined. We also demonstrate how the required products can be defined in the case that singular term is a Gaussian random field by probabilistic calculation. Interestingly, although the terms in the zeroth Wiener chaos of regularization approximation are not zero, they converge in suitable weighted Besov spaces, and no renormalization is required. As applications the global well-posedness for a nonlinear kinetic equation with singular coefficients is obtained by the entropy method. Moreover, we also solve the martingale problem for nonlinear kinetic distribution dependent stochastic differential equations with singular drifts.

Funding Statement

X. Zhang is partially supported by NSFC (No. 12131019, 11731009). R.Z. and X.Z. are grateful to the financial supports by National Key R&D Program of China (No. 2022YFA1006300). R.Z. gratefully acknowledges financial support from the NSFC (No. 12271030) and BIT Science and Technology Innovation Program Project 2022CX01001. X.Z. is partially supported by National Key R&D Program of China (No. 2020YFA0712700), the NSFC (No. 12090014, 12288201) and the support by key Lab of Random Complex Structures and Data Science, Youth Innovation Promotion Association (2020003), Chinese Academy of Science. This work is also funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Collaborative Research Centre (CRC) 1283/2 2021 – 317210226 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications.”


Download Citation

Zimo Hao. Xicheng Zhang. Rongchan Zhu. Xiangchan Zhu. "Singular kinetic equations and applications." Ann. Probab. 52 (2) 576 - 657, March 2024.


Received: 1 August 2021; Revised: 1 September 2023; Published: March 2024
First available in Project Euclid: 4 March 2024

MathSciNet: MR4718402
Digital Object Identifier: 10.1214/23-AOP1666

Primary: 60H15
Secondary: 35R60

Keywords: distributional drift , kinetic equations , Paracontrolled calculus , second-order mean-field SDE , weighted anisotropic Besov spaces

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 2 • March 2024
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